Chap 5. 이항분포와 정규분포


5.1 이항분포 (The Binomial Probability Distribution)
  • Bernoulli experiment : 동전던지기와 같이, 두가지의 결과(s, f)가 가능한 실험
  • Bernoulli random variable : Bernoulli experiment 에서 X(s)=1, X(f)=0으로 정의된 확률변수 X
  • Def. Bernoulli distribution
  • Def. Binomial distribution 성공률이 p인 Bernoulli experiment을 n번 독립적으로 반복할때, 나타나는 성공횟수 X의 확률함수
      1. The experiment consists of a sequence of n trials, where n is fixed in advance of the experiment.
      2. The trials are identical, and each trial can result in one of the same two possible outcomes, which we denote by success(s) or failure(f).
      3. The trials are independent, so that the outcome on any particular trial does not influence the outcome on any other trial.
      4. The probability of success is constant from trial: we denote this probability by p

      　

  • Not. Because the p.m.f of a binomial r.v. X depends on the two parameters n and p, we denote the p.m.f. by b(x;n,p)
  • Thm
  • Not. For X~Bin(n,p) the c.d.f will be denoted by P(X<=x)=B(x;n,p)=\sum_{y=0}^xb(y;n,p)
  • Prop. If X~Bin(n,p) then E(X)=np, Var(X)=np(1-p)

5.2 정규분포 (The Normal Distribution)
  • Def. A continuous r.v. X is said to have a normal distribution with parameter mean \mu and variance \sigma^2, if the p.d.f. of X is
  • Def. The standard normal distribution (\mu=0, \sigma^2=1)
  • Not. z_\alpha will denote the value on the measurement axis for which \alpha of the area under the z curve lies to the right of z_\alpha P(Z>=z_\alpha)=\alpha
  • Prop.If X has a normal distribution with mean \mu and variance \sigma^2, then Z=(X-\mu)/\sigma is a standard normal random variable

5.3 이항분포의 정규근사
  • Prop. The Normal Approximation to the Binomial Distribution
    Let X be a binomial random variable based on n trials with success probability p. Then if the binomial probability histogram is not too skewed, X has approximately a normal distribution with \mu=np and \sigma^2=np(1-p).

Quiz 3. Email: lbg@kowon.dongseo.ac.kr
    1. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked $5 each. Let a person choose, at random and without replacement, 3 chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation.

    2. Let Y be the number of success in n independent repetitions of a random experiment having probability of success p=2/3. If n = 3, compute Pr(2<=Y); if n = 5, compute Pr(3<=Y).

    3. Let Y be the number of success throughout n independent repetitions of a random experiment having probability of success p=1/4. Determine the smallest value of n so that Pr(1<=Y)>=0.7.

    4. If X is n(75,100), find Pr(X < 60) and Pr(70 < X < 100).

    5. If X is n(1,4), compute the probability Pr(1 < X^2 < 9).

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